The Eigenvalue And Pinching Theorem Of Extremal Submanifold

We mainly studies the eigenvalue and rigidity theorem of extremal submanifoldsin a unit sphere in this paper.Let Mⁿ be a closed submanifold in the unit sphereSn p, We call x:M n S^n+p extremal submanifold if it is an critical point to the functional F（x）（see（1.19））.In1968, J. Simons [14] studied geometry classification of n-dimensional compactminimal submanifolds in a unit sphere and he proved the famous Simons rigiditytheorem. In1970, Chern-do Carmo-Kobayashi [22] further given geometryclassification of an n-dimensional compact minimal submanifolds in a sphere S^n+p（1）under the Pinching condition S≤n/（2-1/p）.In2007, Zhen Guo and Hai Zhong Li [4]given the Pinching condition ρ²≤n/（2-1/p） of extremal submanifolds.They also gotsome rigidity theorems. In2011, Deng-yun Yang [5] researched the eigenvalues andrigidity theorem of Willmore submanifolds and Willmore hypersurfaces. Inspired bythem, we study the first eigenvalue of extremal submanifolds in the first part of thisarticle. We estimate the first eigenvalue and its upper bound of the Schr dingeroperator L=-△-（2-1/p）ρ²by selecting the appropriate eigenfunction, and givesome characteristics of submanifold from the angle of eigenvalues.In1989, Chun Li Shen [13] studied a global Pinching theorem of minimalhypersurfaces under the Pinching condition in the unit sphere. In2011, Yang Dengyun[20] studied the gap phenomenon for extremal submanifolds in a sphere. Inspired bythem, we use the Sobolev inequality （2.2） obtained by P. Li in the second part of thepaper and we obtain a global Pinching theorem of another form for compact extremalsubmanifold, where the Pinching constant depends on n, p and M.